Informally/conversationally: I run faster than him would be the most widely used form.

However, I run faster than he does would be both 'safer' in terms of acceptability for more formal purposes while also not out of place in conversation.

I run faster than he, on the other hand (with ellipsis of the verb as opposed to its replacement by a pro-form) is considered by many to be hyperformal and old-fashioned, although one occasionally comes across it in the speech and writing of older people.